Wally Venable:
You asked "what type of tuner?" I just looked at your photo, and you tuners are interesting in their own right due to the way the wood is held between metal fingers. I can't find any match on the Elderly Instrument tuners web page, and there are a lot of designs there.
https://www.elderly.com/pages/search-results?offset=456&q=tuners
Friction pegs (non-violin type) are common on banjos and ukuleles. Looking at the following web pages I identified about five basic types. Some use fiber washers while others use plastic. Yours appear to be fiber since some are more compressed than others.
All About Ukulele Friction Tuners - Got A Ukulele Beginners Tips
Banjo Tuner Tips & Tricks
Some really old high quality instruments used tapered tuning pegs, apparently designed to fit in the holes used for violin pegs. A video dealing with these is:
1920’s Martin Ukulele- What to do about the friction pegs?
One of the videos mentions screws bottoming out. That might be your problem if the wood has shrunk over the past 40 years. That could explain the left-right difference.
Dear Wally, and you are a dear! I am amazed by all the learning I have gotten from your email and those of others who have replied. I never dreamed I would have the courage to take apart the tuners on a musical instrument because, in the past, I have had a 98% failure rate on putting things together that I have taken apart. I didn't even have the language to describe those pegs because I knew they weren't the violin friction type, nor the geared type. So, thank you. As it turns out, they have two metal washers sandwiched between a plastic ring. By taking them apart, and putting them back together, voila, now they work--not sure why except maybe the dulcimer was happier that someone was paying attention to it. So again, many thanks. Linda